We now turn our attention to generic (non-quadratic) continuous differentiable functions \(f\) .
As the expression (REF ) is not suitable since the Hessian is usually not available, we use the generalization (REF ). Just as for the quadratic case (see Proposition REF ), the well-known BB1 and BB2 stepsizes are retrieved for \(\tau _k \rightarrow \pm \infty \)  and \(\tau _k = 0\) , respectively.
In the quadratic case, the secant equation \(\mathbf {y}_{k-1} = B_k \mathbf {s}_{k-1}\)  holds for \(B_k \equiv A\) , and this allows for the interpretation of BB steps and the TBB step as Rayleigh quotients of \(A\) .
When \(f\)  is a generic function, the average Hessian \(B_k = \int _0^1\nabla ^2f(\mathbf {x}_k + t \mathbf {s}_k)\,dt\)  satisfies the secant equation (cf., e.g., [1]}), and thus we can still think at BB steps and the TBB step as Rayleigh quotients, which approximate the eigenvalues of this average \(B_k\)  instead of the ones of \(\nabla ^2f(\mathbf {x}_k)\) .
We note that, under the condition that \(B_k\)  is SPD, all results of Sections  and  continue to hold for generic functions when replacing \(A\)  by \(B_k\) .
